Dynamic response of mesoscopic metal rings and thermodynamics at constant particle number

TL;DR

This paper demonstrates that the thermodynamic persistent current in mesoscopic metal rings at constant particle number aligns with the dynamic current in the zero-frequency limit, revealing fundamental differences from thermodynamic derivatives and limitations in using dynamics to simplify calculations.

Contribution

It establishes the equivalence of thermodynamic and dynamic persistent currents beyond linear response and highlights the challenges in relating dynamic response to Green's functions at fixed particle number.

Findings

Thermodynamic and dynamic persistent currents agree beyond linear response.

Dynamic response involving complex Green's functions cannot be simplified at fixed particle number.

Zero-frequency dynamic current differs fundamentally from thermodynamic derivatives.

Abstract

We show by means of simple exact manipulations that the thermodynamic persistent current $I ( \phi , N )$ in a mesoscopic metal ring threaded by a magnetic flux $\phi$ at constant particle number $N$ agrees even beyond linear response with the dynamic current $I_{dy} ( \phi , N )$ that is defined via the response to a time-dependent flux in the limit that the frequency of the flux vanishes. However, it is impossible to express the disorder average of $I_{dy} ( \phi , N )$ in terms of conventional Green's functions at flux-independent chemical potential, because the part of the dynamic response function that involves two retarded and two advanced Green's functions is not negligible. Therefore the dynamics cannot be used to map a canonical average onto a more tractable grand canonical one. We also calculate the zero frequency limit of the dynamic current at constant chemical potential beyond linear response and show that it is fundamentally different from any thermodynamic derivative.